controlling spatiotemporal pattern formation in a …bib_6cd3635a3577...the boundary can be...

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Controlling spatiotemporal pattern formation in a

concentration gradient with a synthetic toggle switch

Içvara Barbier1, Rubén Perez Carrasco2* and Yolanda Schaerli1*

1 Department of Fundamental Microbiology, University of Lausanne, Biophore Building, 1015 Lausanne,

Switzerland 2 Department of Mathematics, University College London, Gower Street, WC1E 6BT London, United

Kingdom

*Correspondence: [email protected] and [email protected]

Abstract

The formation of spatiotemporal patterns of gene expression is frequently guided by gradients of

diffusible signaling molecules. The toggle switch subnetwork, composed of two cross-repressing

transcription factors, is a common component of gene regulatory networks in charge of patterning,

converting the continuous information provided by the gradient into discrete abutting stripes of gene

expression. We present a synthetic biology framework to understand and characterize the

spatiotemporal patterning properties of the toggle switch. To this end, we built a synthetic toggle switch

controllable by diffusible molecules in Escherichia coli. We analyzed the patterning capabilities of the

circuit by combining quantitative measurements with a mathematical reconstruction of the underlying

dynamical system. The toggle switch can produce robust patterns with sharp boundaries, governed by

bistability and hysteresis. We further demonstrate how the hysteresis, position, timing, and precision of

the boundary can be controlled, highlighting the dynamical flexibility of the circuit.

Keywords: Synthetic biology, gene regulatory networks, pattern formation, dynamical systems,

bistability, hysteresis

.CC-BY-NC 4.0 International licenseIt is made available under a (which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint. http://dx.doi.org/10.1101/849711doi: bioRxiv preprint first posted online Nov. 21, 2019;

http://dx.doi.org/10.1101/849711

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Introduction

Synthetic biology aims to engineer living organisms with standardized and modular circuits that perform

their functions in a programmable and predictable way (Brophy and Voigt, 2014, Cameron et al., 2014,

Purcell and Lu, 2014). In addition to the promise of providing new technologies for medical and industrial

applications (Gilbert and Ellis, 2018, Kitada et al., 2018, Nielsen and Keasling, 2016, Xie and

Fussenegger, 2018), recapitulating biological processes synthetically provides a route to understanding

the basic necessary mechanisms underpinning biological functions and dissect their properties and

limitations (Bashor and Collins, 2018, Li et al., 2018).

Formation of spatiotemporal patterns of gene expression, a crucial process during the development of

multicellular organisms, lends itself to be studied by such a synthetic biology approach. During

development, pattern formation is achieved through a set of inter-connected gene regulatory programs

encoding different non-linear responses to spatial chemical cues. This multiscale complexity makes the

elucidation of the core principles of spatial patterning very challenging in living embryos, calling for

alternative approaches capable of interrogating and comparing different pattern formation mechanisms.

The rise of synthetic biology has successfully allowed to build synthetic systems able to explore core

patterning principles (reviewed in (Santos-Moreno and Schaerli, 2019b, Luo et al., 2019, Davies, 2017,

Ebrahimkhani and Ebisuya, 2019)). In addition, synthetic pattern formation is also an attractive

technology for the engineering of living materials (Gilbert and Ellis, 2018, Nguyen et al., 2018, Moser et

al., 2019, Cao et al., 2017) and tissues (Davies and Cachat, 2016, Healy and Deans, 2019, Webster et

al., 2016).

One ubiquitous strategy of patterning during embryogenesis is positional information, in which signaling

molecules – the morphogens – diffuse and generate concentration gradients to specify positions.

Specific gene regulatory programs are able to translate the spatiotemporal information provided by the

local concentration of morphogen gradients into robust gene expression patterns (Wolpert, 1996, Green

and Sharpe, 2015, Rogers and Schier, 2011). This mechanism has been extensively used in synthetic

systems to generate spatial patterns, especially stripe patterns, which were produced through synthetic

feed-forward loops (Basu et al., 2005, Schaerli et al., 2014), inducible promoters (Grant et al., 2016) and

AND-gates (Boehm et al., 2018).

One of the gene regulatory subnetworks able of interpreting positional information is the bistable genetic

switch (Zhang et al., 2012, Srinivasan et al., 2014, Balaskas et al., 2012, Kraut and Levine, 1991, Lopes

et al., 2008b, Perez-Carrasco et al., 2016, Sokolowski et al., 2012, Zagorski et al., 2017, Alon, 2007),

known as toggle switch (TS) (Figure 1A). The topology of this circuit consists of two cross-repressing

nodes that result in the binary mutually exclusive stable expression of one of the nodes. If the expression

is influenced by an external signal, the TS provides a mechanism to convert a concentration gradient of

this signal into stripes of gene expression (Perez-Carrasco et al., 2016, Sokolowski et al., 2012).

Examples of TS-controlled pattern formation have been identified in the mesoderm formation in Xenopus

(Saka and Smith, 2007), Drosophila blastoderm gap gene segmentation (Verd et al., 2019, Clark, 2017),



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and neural specification in vertebrate neural tube (Briscoe and Small, 2015, Perez-Carrasco et al.,

2018).

The non-linearity of gene regulatory networks such as the TS impedes to intuitively understand the effect

that different kinetic parameters have on the dynamics of gene expression. For this reason, during the

last decade, gene regulatory networks have been analyzed using tools from dynamical systems theory,

associating the stable steady states of the dynamical system with the attainable gene expression states

of a cell. Similarly, the change in the availability of cellular states as a consequence of perturbations of

kinetic parameters of the network can be associated with the bifurcations of the dynamical system,

providing information of the constraints of the possible cellular states. Indeed, the dynamical system of

the TS has been thoroughly analyzed in silico, both in single cells as well as in population-level

patterning scenarios (Perez-Carrasco et al., 2016, Ferrell, 2002, Guantes and Poyatos, 2008), showing

that two possible stable states can coexist for a region of parameters inside which the expression state

of the cell will depend on the initial gene expression – a phenomenon known as bistability. Under the

control of an external signal, this bistability leads to hysteresis in which the state of the system is robust

to signal changes, thus providing memory to gene expression patterns (Wang et al., 2009). Interestingly,

the analysis of the steady states of the dynamical system not only provides static information of the

cellular states, but also information on the transient dynamics of gene expression by which the states

are attained (Verd et al., 2014). Therefore, a map of the underlying dynamical system is critical to fully

understand the dynamics of the TS network.

The first synthetic version of the TS network was built almost 20 years ago and was a milestone of

synthetic biology (Gardner et al., 2000). Since then, it has been built multiple times, extensively studied

and used for its memory, bistability or hysteresis properties (Padirac et al., 2012, Purcell and Lu, 2014,

Chen and Arkin, 2012, Lou et al., 2010, Andrews et al., 2018, Yang et al., 2019, Nikolaev and Sontag,

2016, Zhao et al., 2015, Sokolowski et al., 2012, Lebar et al., 2014), for stochasticity fate choice (Axelrod

et al., 2015, Perez-Carrasco et al., 2016, Sekine et al., 2011, Wu et al., 2013, Lugagne et al., 2017) and

to tune threshold activation (Gao et al., 2018). Nevertheless, its patterning capabilities controlled with a

morphogen-like signal have not been studied in a synthetic system. Here, we constructed a

“morphogen”-inducible synthetic TS network and studied its ability to produce spatial patterns - governed

by bistability and hysteresis - in an Escherichia coli (E. coli) population. A combination of experiments

and mathematical modelling allowed us to characterize the underlying bifurcation diagram, unveiling the

possible dynamical regimes of the circuit. This enabled us to demonstrate how the inducible TS allows

to control hysteresis, precision, position and timing of the pattern boundary.



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Results

Inducible toggle switch topology and its spatial patterning behavior.

The inducible TS network consists of two mutually repressing nodes (Figure 1A). This mutual inhibition

architecture ensures that only one of the nodes can be maintained at high expression. The dichotomous

response of the cell depends on the asymmetry of the repression strengths and on the production rates

of each node. An array of cells under a concentration gradient in charge of controlling any of these two

properties will generate a binary spatial pattern. We built a synthetic inducible TS circuit starting from a

characterized TS (Litcofsky et al., 2012). The first node of the network is composed of the TetR repressor

and the mCherry reporter and will be referred hereon as the red node. The second node contains the

LacI repressor and GFP reporter and will be referred as green node (Figure 1B). Accordingly, the two

expression states that the circuit can maintain will be referred as green and red states. TetR and

mCherry are regulated by the hybrid pLuxLac promoter (BBa_I751502), which is activated by the LuxR-

AHL (N-(β-Ketocaproyl)-L-homoserine lactone) complex and repressed by LacI, whose repression

strength can be regulated by isopropyl β-d-1-thiogalactopyranoside (IPTG). LacI and GFP are controlled

by the pLtetO promoter, which is repressed by TetR. LuxR is constitutively expressed from a pLuxL

promoter on a second plasmid (Figure 1B). In the absence of AHL (inducer), TetR and mCherry are not

expressed, but LacI and GFP are, resulting in the green state. In the presence of AHL, TetR and

mCherry can be expressed and repress LacI and GFP expression. Consequently, the system switches

to the red state, provided that the concentration of the regulator (IPTG) is high enough.

We studied the capability of the inducible TS circuit to pattern an E. coli population exposed to different

concentrations of AHL and IPTG and combinations thereof (Figure 1C-D). To this end, we grew the cells

on a hydrophobic grid placed on an agar plate to give a defined spatial organization to the cells (Grant

et al., 2016), while small molecules can freely diffuse between grid squares. AHL and IPTG were

pipetted at the left and at the top edges of the grid, respectively, forming gradients of AHL and IPTG by

diffusion. We then measured the bacterial fluorescence after overnight incubation. When the starting

cells were in the green state (reached by previous incubation in the absence of AHL and IPTG), the

switch to the red state was observed only in presence of both AHL and IPTG, in the top left corner of

the grid (Figure 1C). In contrast, the same experiment performed with cells initially in the red state

(reached by previous incubation in the presence of AHL and IPTG), showed that the red state is

maintained above a certain concentration of AHL, mostly independent of the concentration of IPTG,

leading to a green domain at the right (Figure 1D). An overlay of the two grid patterns allows to highlight

the possible stable states available for different concentrations of AHL and IPTG showing two

monostable regions for the red and green states, and a bistable region for low values of AHL and high

values of IPTG that grant hysteresis to the system (Figure 1E). In addition to the tuning of the LacI

repression by IPTG as explored here, the TetR repressing strength can be regulated by the addition of

anhydrotetracycline (aTc) (Supplementary Figure 1A). Different combinations of gradients of AHL, IPTG

and aTc result in different patterns (Supplementary Figure 1B-D), revealing the flexibility of the TS to

control spatial gene expression.



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Figure 1: Inducible toggle switch topology and its spatial patterning behavior.

A. Schematic of the inducible toggle switch network composed of two mutually inhibitory nodes. The expression of the red node

is controlled by an inducer, while a regulatory molecule can be added to tune the repression strength of the green node. B.

Detailed representation of the molecular implementation of the network in (A) using SBOL annotation (Beal et al., 2019). C-D. 2D

spatial patterning of a population of E. coli cells harboring the inducible toggle switch network. The colors correspond to the levels

of fluorescence of mCherry (red) or GFP (green) produced by bacteria grown on a grid. 5 µl of aqueous solutions of 100 mM IPTG

and 100 μM AHL were added at the grid edges forming gradients by diffusion as indicated by the triangle shapes. Before growing

on the grid, bacteria were turned into the green (C) or red state (D) as indicated at the bottom. The grids were imaged after

overnight incubation. Each grid square measures 0.75 x 0.75 mm2. E. Overlay of the grids of C and D, highlighting the hysteresis

of the system in yellow, resulting from the superposition of red and green fluorescence.

Characterization of the toggle switch as a dynamical system

To quantitatively analyze the network behavior with single cell resolution, we grew the TS bacteria in

liquid culture for 10 h with defined concentrations of inducer (AHL) and regulator (IPTG) molecules and

used flow cytometry to measure the green and red fluorescence of individual bacteria (Figure 2A). We

gated the bacteria to quantify the percentage of cells in the red or green state for different inducer and

regulator concentrations and different initial states. The observed behavior is consistent with the spatial

patterning on the grid: Starting with an initial population of bacteria in the green state, the entire

population (>90% of events) switched to the red state at high concentrations of both AHL and IPTG,

whereas they stayed green otherwise. On the other hand, starting in the red state, the entire population

stayed red above AHL concentrations of ~11 nM, but switched to the green state at lower AHL

concentrations.



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At the boundary between the red and green areas (white squares in Figure 2A), we observed cells in

both flow cytometry gates. At high IPTG concentrations (≥0.5mM), a single population was located at

intermediate red and green fluorescence values, whereas at IPTG concentrations ≤0.5mM the

population split into two subpopulations displaying either the red or the green state. A bimodal

distribution at the boundary between the two stable states is a known phenomenon of bistable circuits

(Axelrod et al., 2015, Wu et al., 2013, Sekine et al., 2011) and can be caused by cell state switching in

response to intrinsic gene expression noise (Perez-Carrasco et al., 2016, Sekine et al., 2011).

The transition from red to green is less affected by the concentrations of IPTG than the transition from

green to red, because of the asymmetry of the network. In the transition from red to green, the rate-

limiting step is the degradation and dilution of TetR in the cells, since the clearance of TetR is required

before changes in AHL or IPTG concentrations can induce the switch. Therefore, for the rest of the

manuscript, we will focus on initial green state populations, while the data for the initial red state is in

the supplementary data (Supplementary Figures 4 and 5).

In order to fully characterize the dynamical capabilities of the circuit, we described the network with a

mathematical model composed of ordinary differential equations (ODE) capturing the concentration of

all the chemical species over time (see Methods for details). To unveil the dynamical landscape

compatible with the synthetic circuit, we parameterized the model by fitting it to our experimental data.

In order to do so, we made use of multitry Markov chain Monte Carlo (MCMC) inference (Laloy and

Vrugt, 2012, Shockley et al., 2018), with a likelihood function based on the experimentally measured

levels of expression and number of steady states found for the different concentrations of AHL and IPTG

tested. We obtained a narrow probability distribution for all 11 parameters (Supplementary Figure 2).

The predicted outputs from the parametrized equation were able to recapitulate the bistability and

hysteresis observed in the experimental data (Figure 2B and Supplementary Figure 3). This allowed us

to reconstruct the multidimensional bifurcation diagram of the system (Figure 2C) which we can use as

a map to predict the effect that different dynamical gradients will have on the observed gene expression

patterns.

Analysis of the bifurcations of the system shows a scenario compatible with other TS, in which a

continuous variation of AHL or IPTG can change the number of stable states via saddle-node

bifurcations. Starting from a bistable condition, a saddle-node bifurcation occurs through the collision of

a stable and an unstable state of the system, leaving only one possible stable state left (Figure 2C). For

a cell in the state about to disappear, a small perturbation of IPTG or AHL induces a sharp switch to the

opposite state, producing a sharp transition between cell states.

There are two different saddle-node lines in the phase plane (AHL, IPTG) of the bifurcation diagram that

collide at a certain concentration of the inducers (around 10-3 µM AHL and 10 mM IPTG). Called a cusp

bifurcation, this point separates the regions of inducer in the phase plane in which the switch between

states occur through a saddle-node bifurcation (bistability) or through the continuous change of



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expression of a monostable state. Thus, the availability of a cusp point allows to explore the properties

of two different patterning strategies (bistability versus sigmoidal response) for the same circuit topology.

Controlling the hysteresis, position, and sharpness of the boundary

Next, we analyzed how the IPTG concentration influences the hysteresis of the circuit, characterized by

the range of inducer concentrations for which the circuit shows bistability (Figure 2D). Since IPTG is in

control of the repression of the green node over the red node, it is a perfect candidate to control the

hysteresis of the TS. Based on the bifurcation diagram, we expected that the lower the IPTG

concentration (i.e. the stronger the repression on the red node), the bigger the range of bistability.

Testing the bistability of the circuit at different IPTG concentrations confirmed this hypothesis, showing

a response without bistability to an AHL gradient at high values of IPTG (10 mM) and an increasing

range of bistability as IPTG decreases. The highest amount of hysteresis is observed in the absence of

IPTG. Here, cells are not able to switch from green to red, even in presence of high AHL. However, AHL

is enough to preserve the red state once reached. We thus observed three different dynamic regimes

with distinct ranges of hysteresis: 1) no hysteresis (>1 mM IPTG) in a sigmoidal regime. 2) hysteresis,

in a bistable regime with the possibility to switch between both states (around 0.125 mM IPTG) and 3)

hysteresis in a bistable regime with irreversibility of the green state (around 0 mM IPTG). Those three

regimes were also observed as spatial patterns on the grid, when placed on agar plates with uniform

IPTG concentrations (Figure 2D, right).

In addition to controlling the transition between a sigmoidal and a bistable regime and consequently the

hysteresis, the IPTG concentration also affects the position of the boundary between the red and green

states. Higher concentrations of IPTG allow the system to switch from green to red at lower AHL

concentrations (Figure 2A and D), thus controlling the boundary position for a given AHL gradient.

Finally, IPTG also tunes the transition sharpness between the red and green states (Figure 2E and

Supplementary Figure 4). Starting from the green state, at high concentrations of IPTG beyond the cusp

point, we observed a sigmoidal expression response in an AHL concentration gradient, similar to the

one expected from a circuit with a single repressor (TetR) active. At lower IPTG concentrations, below

the cusp point, the system displayed a sharper transition as expected from the saddle-node bifurcation.

Moreover, these different transitions are consistent with the distinct population behaviors during cell

state transitions along the gradient: a smooth transition of a single population for the sigmoidal behavior

and a bimodal population transition (Figure 2A) for the bistable switch behavior. In summary, the

regulator IPTG allows us to choose between a bistable and a sigmoidal regime (at high IPTG

concentrations) and thus to control the hysteresis, the position and the sharpness of the boundary.



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Figure 2: Controlling the hysteresis, position, and sharpness of the boundary. A. Quantitative single cell analysis of the inducible toggle switch. Overview of the flow cytometry data of a representative replicate.

Each square shows red (Y-axis) and green (X-axis) fluorescence of the population (10’000 events) measured at indicated

concentrations of IPTG and AHL after 10 h of incubation. Background color corresponds to the percentage of red gated events

as indicated. B. Comparison between the observed populated states from the flow cytometry data (circles) and the available

steady states predicted by the model (solid lines: stable states, dashed lines: unstable states), shown for 3 IPTG concentrations.

The median and the standard deviation of the experimentally observed states of the gated populations are shown. Parameters

used in the model are the best parameter candidates from the MCMC fitting (Table 1). The whole data set is shown in

Supplementary Figure 3. C. Top: Bifurcation diagram for the parameters found in the fitting, showing the stable steady states

available for different combinations of [AHL] and [IPTG]. Bottom: Phase portrait of the TS, indicating monostable regions in

red/green proportionally to the concentrations of mCherry/GFP and bistable conditions in blue. The circle indicates the cusp

bifurcation. D. Hysteresis at three different IPTG concentrations, corresponding to the three different regimes showed in (C) as



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dashed lines. Left) Mean percentage of red cells for different concentrations of AHL are shown for an initial population in the red

state (solid red lines) and for an initial population in the green state (dashed orange lines). Error bars show standard deviation of

3 biological replicates. Right) Grid patterns at constant IPTG concentrations. 5 µl of a solution of 100 µM AHL were added at the

left to create the gradient. Colors correspond to the intensity of red and green fluorescence. E. Sharpness of the boundary.

Normalized red fluorescence for an initial population in the green state in an AHL gradient at different IPTG concentrations. Mean

and standard deviation of 3 biological replicates. The inset bar plots represent the maximum slope of each curve (Supplementary

Figure 3).

Temporal dynamics of the patterning

So far, we investigated the influence of IPTG on the pattern in an AHL gradient by studying the static

spatial gene expression after overnight incubation. However, in order to understand the process of

pattern formation it is of paramount importance to study the temporal dynamics of the patterning

process. To this end, we measured the fluorescence of the cells at different concentrations of IPTG and

AHL over a time course of 10 h with flow cytometry (Figure 3, Supplementary Movie 1, Supplementary

Figure 5). We observed that gradients inducing a pattern across a bistable region have a slower and

position-dependent response than those patterning across a sigmoidal region. In particular, the time to

switch from green to red (>90% red events) depends on the concentration of IPTG, switching at 4 h at

high IPTG concentrations (10 to 0.5 mM) and after 6 h and 8 h for lower IPTG concentrations (0.16 and

0.125 mM, respectively) (Figure 3A). Similarly, for a constant amount of IPTG, the switching time

depends on the AHL concentration, switching at times as slow as 6-10 hours for an IPTG concentration

corresponding to the bistable region (0.125 mM Figure 3B). On the other hand, inducing a pattern in the

sigmoidal regime (10 mM IPTG), we observe a more consistent switching time (4 h) across different

AHL concentrations (Figure 3C).

Interestingly, the behaviors of the switch at the population level over time are identical to the ones

observed across different AHL concentrations: at high IPTG (beyond the cusp bifurcation, in the

sigmoidal regime) we measured one population moving as a whole, while at lower IPTG concentrations

(bistable regime) we observed the cells splitting into two divergent subpopulations (Figure 3A,B and

Supplementary Movie 1), suggesting that the dynamics of the sigmoidal patterning are a result of the

population relaxing to the unique possible steady state, whereas in the bistable regime the transition is

controlled through the stochastic switching between the two possible states, with a time that is

determined by the intrinsic noise and can therefore be slower than the degradation rate of the proteins

composing the TS.



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Figure 3: Temporal dynamics of patterning with the inducible toggle switch. A-C. Effects of IPTG and AHL concentrations on the switching time from the green to the red state. For plots in the right column,

each square represents flow cytometer data of 10,000 events measuring red (Y-axis) and green fluorescence (X-axis).

Background color of each square indicates if >90% of the events are in the red or the green gate. Plots in the left column show

the percentage of cells in the red gate over time. Points indicate the mean and standard deviation of 3 biological replicates. Dashed

lines indicate the 90% threshold used to color the flow cytometry plots. A. Analysis of state transition over time in the presence of

different IPTG concentrations and a high level of AHL (10 µM). B-C. Analysis of state transition over time in the presence of

different AHL concentrations and two different IPTG concentrations corresponding to the sigmoidal (10 mM) and bistable regimes

(0.125 mM).

Patterning in the sigmoidal regime is faster than in the bistable regime

Combining our model with 2D diffusion allowed us to reproduce the patterns observed in the grid assay

shown in Figure 1 (Figure 4A). In addition, integrating the diffusion of the inducer with the dynamical

properties of the bifurcations of the system can shed light on the different patterning dynamics observed.

In particular, consistent with our flow cytometry data (Figure 3), the model suggests that the switching

slows down close to the saddle node bifurcation, a phenomenon called critical slow down. Thus, for

different constant values of IPTG, a gradient of AHL is expected to create a moving boundary at different



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speeds for cells switching from the green to the red state (Figure 4B) (Perez-Carrasco et al., 2016). To

test this prediction, we measured the position of the boundary over time in the grid assay at two different

IPTG concentrations corresponding to two different dynamical regimes (sigmoidal, 1 mM and bistable,

0.125 mM) (Figure 4B). As expected, we observed that the transition to the production of mCherry starts

earlier and advances faster in the sigmoidal regime, equipping the TS with time control of the pattern

formation through the cusp bifurcation. This was confirmed through simulation of the diffusion of the

inducers on the grid, were the model predicts the same trend than the experimental data, with no more

than three squares of difference. On the other hand, we did not observe clear differences in the precision

of the boundary in the grid assay, probably due to a lack of precision in the concentration gradients

accessible by the grid.

Figure 4: Patterning in the sigmoidal regime is faster than in the bistable regime

A. Modelling of the pattern observed in the grid assay (Figure 1C-E). Top left: AHL diffusion from left to right and IPTG diffusion

from top to bottom. Bottom left and right: Colors represent the state of the system, green for GFP/LacI expression, red for

mCherry/TetR expression, and yellow when both states are possible depending on the initial state. B. Time course of pattern

formation in the grid in the sigmoidal (1 mM IPTG, left) and bistable (0.125 mM, right) regimes. Center: Before growing on the

grid, bacteria were turned into the green state. The indicated concentration of IPTG was homogeneously present in the agar plate

and 5 µl of 100 µM AHL were loaded at the left edge of the grid. The grids were incubated at 37° C and imaged at the indicated

times. A representative replicate is shown. Left and right 3D plots represent the quantitative analysis of mean red fluorescence

intensity in the grid over time and position of 3 biological replicates. Blue lines represent the boundary predictions from the

mathematical model. Green and red colors correspond to measurements where the red fluorescence intensity is below or above

50% of the highest intensity measured along the AHL gradient at each timepoint, respectively, corresponding to how the boundary

was defined in silico (see Methods for details).



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Spatiotemporal control of the pattern by using spatially homogenous signals

In addition to the pattern formation through a bistable or monostable sigmoidal regime, the TS offers the

possibility to move between both different regimes during the pattern formation, allowing for different

patterning strategies that can exploit the properties of both regimes. In particular, by manipulating the

homogeneous levels of IPTG in time for a given gradient of AHL, we can control the pattern formation

process. An initial population of cells in the green state in the absence of IPTG (bistable irreversible

regime) is unaffected by the gradient of AHL. Adding IPTG homogeneously at the desired time point,

brings the cells to the sigmoidal monostable regime able to respond to the gradient of AHL and forming

a boundary. Once the boundary is located at the desired position, removing IPTG takes the system back

to the bistable zone, thus freezing the boundary and making it robust to changes in AHL (Figure 5A).

Consequently, the system is able to maintain a pattern in the absence of IPTG which removes the

requirement of maintaining a precise AHL gradient to keep the pattern boundary. Therefore, a pulse of

IPTG can combine advantages of two distinct regimes: of the sigmoidal monostable regime for a fast

establishment of a pattern (Figure 4) and of the irreversible regime to make the pattern robust to changes

in the AHL gradient. To demonstrate this effect, we grew cells initially in the green state at different

concentrations of AHL and exposed them to pulses (2 h, 3.5 h ,5 h, 6.5 h) of 10 mM IPTG (Figure 5B).

We then grew them for further 6 h in the absence of IPTG, but keeping the AHL concentrations used

during the pulse. Indeed, cells receiving enough AHL (>= 0.01 µM), were able to maintain the red state.

Interestingly, this was the case even for short pulses were not the whole population did have enough

time to switch (2 h). For intermediate values of AHL, close to the boundary, the final state of the

population depends on the position of the population with respect to the basins of attraction of the

bistable regime. In particular, it is noteworthy that if the single cell population at the end of the pulse is

located at intermediate green and red fluorescence levels, it splits into two subpopulations (green or red

fluorescence) when brought back to the bistable regime.

We used the grid assay to further demonstrate this memory property of the TS and to show that the

pattern in the bistable regime is indeed robust to changes in the AHL gradient. We patterned a grid as

in Figure 1 and transferred the cells onto a new agar plate where the positional information provided by

AHL and IPTG was removed (0 mM IPTG, homogeneous concentration of 5 µM AHL). As predicted, the

pattern was maintained in the absence of any spatial information (Supplementary Figure 6). This result

demonstrates that an inducible TS network is capable of interpreting spatiotemporal gradients by making

use of the memory properties of the circuit.



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Figure 5: A homogeneous pulse of IPTG allows to control the pattern formation. A. Schematic of the protocol used. Cells

initially in the green state for low levels of IPTG (irreversible bistable regime) are exposed to a pulse of high IPTG (at time 𝑡 )

bringing the system to the sigmoidal region where a gradient of AHL can induce a pattern. Removing IPTG (at time 𝑡 ) takes the

system back to the bistable zone, thus freezing the boundary and making it robust to changes in AHL (at any posterior time 𝑡 ).

B. Response of the system to IPTG pulses of different durations (2- 6.5 h) at different AHL concentrations. Cells were grown in

the presence of 10 mM IPTG and the indicated amount of AHL and analyzed by flow cytometry. The top graph displays the

percentages of red cells observed just after the incubation with IPTG (time 1, 𝑡 ). Next, the cells were diluted and grown for another

6 h without IPTG, but with the same concentration of AHL as during the IPTG pulse. The bottom graph displays the percentage

of red cells observed after this incubation (time 2, 𝑡 ). Inset graphs represent red (y-axis) and green (x-axis) fluorescence measured

by flow cytometry (10,000 events). The following conditions are shown: (1) 6.5 h pulse, 1.235 nM AHL. (2) 6.5 h pulse, 10000 nM

AHL. (3) 2 h pulse, 10000 nM AHL.

Discussion

Systems displaying hysteresis, and in particular the bistable switch, allow for a sharp threshold response

that can turn a graded input into a binary output (Perez-Carrasco et al., 2016, Sokolowski et al., 2012).

Interestingly, the mutual repressing motif of the toggle switch is widely found in natural pattern-forming

systems, for example, in networks responsible for patterning the neural tube (Balaskas et al., 2012), the

dorsal telencephalon (Srinivasan et al., 2014), the Drosophila embryo segments (Verd et al., 2019,

Clark, 2017) and the Xenopus mesoderm (Saka and Smith, 2007). However, in addition to forming sharp

boundaries, the successful formation of a pattern requires the control of the position and timing of the

boundary formation. In order to explore dynamic patterning properties of the TS gene regulatory

network, we have successfully built and characterized a synthetic inducible toggle switch with tunable

repression (Figure 1).

We quantified the gene expression at the single cell level over a wide range of inducer and regulator

concentrations and fit the data to parameterize a mathematical model of gene regulation. The resulting



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bifurcation diagram of the dynamical system provided the mechanistic understanding required to

interpret the different spatiotemporal patterns observed and experimental design guidance for pattern

formation with the synthetic TS. In particular, this approach allowed us to characterize the mechanism

by which the system can transit from a bistable regime to a sigmoidal unimodal response to the inducer,

determined by the cusp bifurcation of the dynamical system (Figure 2). From the exploration of the

differences in boundary precision of both regimes, we observed that while the bistable regime provides

the sharper boundary (as previously reported (Lopes et al., 2008a, Isalan et al., 2005)), the sigmoidal

regime allows for a faster response (Figures 3 and 4). This reveals a trade-off between the timing and

precision of the boundary. These observations are in consonance with dynamical systems predictions,

in which the sharp transition of the bistable regime - through a saddle-node bifurcation - comes at the

price of the critical slowing down close to the bifurcation (Narula et al., 2013, Perez-Carrasco et al.,

2016).

In addition, for different levels of a spatially homogeneous regulator, we have been able to control the

range of hysteresis and the position of the pattern boundary for a given inducer gradient changed

accordingly (Figure 2). This mechanism is analogue to the one proposed by Cohen et al. (Cohen et al.,

2014) to control the boundary position of patterns regulated by morphogen gradients by changing the

affinity of one of the elements to a spatially homogeneous transcription factor. This result highlights the

evolutionary potential of the circuit, allowing for kinetic mechanisms of controlling the position of the

boundary without compromising other aspects of the boundary or requiring an upstream regulation of

the morphogen gradient.

Further to controlling the position of the pattern, the range of bistability can also lead to an irreversible

switch for a wide range of the parameter space. Inside this region of the parameter space, any prepattern

can be robustly fixed, only requiring the morphogen information for a certain time window (Figure 5,

Supplementary Figure 6). Coupled with the dynamics of boundary formation, this provides alternative

patterning strategies to the classical static positional information, in which the precision, position and

timing of the boundary can be controlled by the dynamic properties of the upstream signal. This flexibility

is of utmost importance in developmental scenarios, where both, the morphogen signalling and the

patterns are dynamic processes during tissue differentiation, such as the boundary position movement

during the gap gene segmentantion in Drosophila (DiFrisco and Jaeger, 2019), or the adaption of the

signalling of sonic hedgehog morphogen during neural differentiation in the patterning of the neural tube

(Balaskas et al., 2012). In this latter scenario, the adapatation to the gradient of sonic hedgehog (through

Gli signalling) results in a combination of temporal profiles of activation and repression acting on the

patterning circuit (Junker et al., 2014, Cohen et al., 2015). This highlights the importance to understand

bistable switches inside a dynamical scenario. While such dynamics is challenging to measure in the

developing embryo, our synthetic biology framework allowed for an alternative route towards

understanding the dynamics of the TS in pattern formation.



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Our improved understanding of the TS for pattern formation opens up new avenues of research. In

particular, for many bistable parameter conditions, single cell expression showed the coexistence of two

subpopulation of cells at each one of the available stable states. This provides evidence of the relevance

of intrinsic noise in the establishment of the pattern. While previous in silico research shows that intrinsic

noise can determine the position and precision of morphogen driven boundaries (Perez-Carrasco et al.,

2016, Weber and Buceta, 2013), the actual role of noise in the dynamics of pattern formation in living

systems and the possibility of optimal dynamical strategies based on the stochasticity of gene

expression remain still a conundrum. In addition, it poses new challenges to dynamical system inference

in which different sources of intrinsic noise must be disentangled from measurement noise in order to

obtain an accurate characterization of the circuit (Dharmarajan et al., 2019).

Overall, our results underscore the relevance of studying the dynamical context of a gene regulatory

network in order to understand patterning processes, not only of synthetic circuits but also of

developmental systems (Ferrell, 2002, Sagner and Briscoe, 2017). Future work will reveal if the TS

properties are conserved when incorporated in more complex (synthetic) gene regulatory networks, for

example when combined with the repressilator (Elowitz and Leibler, 2000, Potvin-Trottier et al., 2016,

Santos-Moreno and Schaerli, 2019a) to yield the AC-DC network (Perez-Carrasco et al., 2018, Verd et

al., 2019, Balaskas et al., 2012, Panovska-Griffiths et al., 2013). Moreover, the here established

engineering guidelines on how to control patterning with a synthetic TS will be valuable for future

synthetic pattern formation, for example in the context of engineered living materials based on bacterial

biofilms (Gilbert and Ellis, 2018, Nguyen et al., 2018, Moser et al., 2019, Cao et al., 2017).



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Materials

Media

Cloning experiments used lysogeny broth medium (LB: 10g Bacto-tryptone, 5 g yeast extract, 10 g NaCl

per 1 l) supplemented with the appropriate antibiotic (25 μg/ml kanamycin or 25 μg/ml spectinomycin).

All experiments with the synthetic circuit were performed in M9 minimal medium (1x M9 salts, 2mM

MgSO4, 0.2mM CaCl2, 0.0005% (w/v) thiamine) with 0.2% (w/v) glucose as carbon source,

supplemented with 0.1% (w/v) casamino acids and the appropriate antibiotics (25 µg/mL kanamycin and

25 μg/ml spectinomycin)

Reagents

Restriction enzymes, alkaline phosphatase from calf intestine (CIP), DNA Polymerase I large (Klenow)

fragment and T4 DNA ligase were purchased from New England BioLabs (NEB). Oligonucleotides and

chemicals were obtained from Sigma-Aldrich. Polymerase chain reactions (PCRs) were carried out with

Q5 Hot Start High-Fidelity DNA polymerase (NEB). Colony PCRs were performed with Taq polymerase

(NEB). PCR products and digested plasmids were purified with the Monarch PCR & DNA Cleanup Kit

(NEB). Plasmids were purified using the QIAprep Spin Miniprep Kit (QIAGEN).

Methods

Cloning of the inducible toggle switch circuit

We cloned the morphogen inducible toggle switch plasmid (“TS_pLuxLac”) from an already functional

toggle switch plasmid (pKDL071) (Litcofsky et al., 2012), kindly provided by Jeong Wook Lee. The two

Ptrc-2 promoters upstream of TetR and mCherry coding sequences were replaced by the hybrid

promoter pLuxLac (BBa_I751502) with the plug-and-play method initially used to assemble pKDL071

using the restrictions sites NcoI and SalI upstream of TetR and XmaI and MfeI upstream of mCherry.

For the “pCDF_luxR” plasmid, the pLuxL promoter (BBa_R0063) and the LuxR gene (synthesized by

GenScript) were introduced into a pCDF plasmid with a customized multiple cloning site (Schaerli et al.,

2014) between the KpnI and BamHI sites. The sequences and plasmids will be available through

Addgene.

Strain and growth condition

pCDF_luxR and TS_pLuxLac were transformed into the E. coli strain MK01 (Kogenaru and Tans, 2014).

The absence of the lacI gene in this strain avoids unexpected cross talk between the synthetic circuit

and the host.

Bacteria were turned into red state by inoculating single colonies into 4 ml M9 medium in presence of 1

mM IPTG (isopropyl β-D-1-thiogalactopyranoside) and 10 µM AHL (N-(β-Ketocaproyl)-L-homoserine

lactone). They were grown overnight at 37° C and 200 RPM shaking. The same procedure was used to

turn the bacteria into the green state, with the difference that the medium did not contain IPTG and AHL.

These bacteria were plated out on LB agar plates (supplemented with 10 µM AHL for the red state) and



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incubated overnight at 37° C. The plates were stored at 4° C and single colony from them were picked

for the experiments.

Flow cytometry

Single colonies in the red or green state were cultured in M9 for 4-6 h and put at 4° C before entering in

stationary phase (below 0.8 OD). The following day, these cultures were diluted to 0.01 OD (NanoDrop

2000, Thermofisher) and added into the wells of a 96-well plate (CytoOne) to a total volume of 120 µL

including indicated concentrations of AHL and IPTG. The plate was incubated in a BioTek Synergy H1

plate reader at 37° C with 548 cpm (2 mm) double orbital shaking speed. Absorbance (600 nm) was

monitored every 10 min to check that cells did not enter stationary phase (below 0.3 in the plate reader)

as cells in stationary phase can no longer switch between the two states. After 5 h, cells were diluted

100 times before incubating them again under the same conditions for a total of 10 h.

At indicated times, 5 µL of the cell cultures were diluted into 95 µL of phosphate-buffered saline (PBS)

and analyzed by flow cytometry (BD LSRFortessa™) with 488 nm excitation and FITC filter for

measuring GFP fluorescence and 561 nm excitation and PE-Texas Red filter for measuring mCherry

fluorescence. 10,000 events were recorded and analyzed by FlowJo and a custom-made R script.

First, cells were gated with FlowJo in the SSC-H and FSC-H scatter plot. Next, FITC-H and PE-Texas

Red-H data were exported to be analyzed in R. The red and green gates were set with the help of

positive controls for red and green fluorescence. These controls were an overnight culture of our bacteria

in presence of 1 mM IPTG and 10 µM AHL for red and without any inducer for green. In the figures,

percentages of red cells correspond to the percentages of cells found in the red gate and the red

fluorescence mean corresponds to the mean of red fluorescence of all the gated cells normalized to the

red fluorescence of the cells grown at the highest AHL and IPTG concentrations and corrected for the

background fluorescence (minimal red fluorescence value measured in the experiment).

For the IPTG time pulse experiment, we started the experiment with a culture of OD 0.1 in a medium

containing 10 mM IPTG and the indicated concentration of AHL. The samples were incubated as

described above until the first time point (2 h) and an aliquot was stored at 4° C. Then, the cells were

diluted 1:5 into fresh medium containing the same IPTG and AHL concentrations and further incubated

until the next time point (3.5 h). This procedure was repeated for all time points. An aliquot of each

sample was analyzed by flow cytometry once all samples were collected. The next day, the cells where

diluted 1’000 times into fresh medium with the indicated concentration of AHL and no IPTG. The cells

were grown for 6 h and directly measured by flow cytometry.

Grid assay

Single colonies in the red or green state were cultured in M9 for 4-6 h and put at 4° C before entering in

stationary phase (below 0.8 OD). The following day, these cultures were washed from IPTG and AHL

by centrifugation for 1 min at 13’000 RPM and resuspended in fresh medium without IPTG or AHL. Then

cells were diluted to an OD of 0.1 before being pipetted onto the grid (ISO-GRID from NEOGEN) (Grant

et al., 2016) (20 μl of cells were loaded for 10 lines, approximatively 0.05 μl per square) that was placed

on a M9 agar plate with 0.2% (w/v) glucose and appropriate antibiotics (25 µg/mL kanamycin and 25



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µg/mL spectinomycin). The inducers (100 mM IPTG and 100 μM AHL, 5 μl each) were added as

indicated in the figures. The plate was incubated overnight at 37° C before green and red fluorescence

were measured with a Fusion FX (VILBER) imaging system. We used 0.2 ms exposure with blue light

(480nm) and a F-565 filter for the GFP measurement and 1 min exposure with red light (530nm) and F-

740 filter for mCherry. ImageJ software (R Core Team, 2017) was used to analyze the picture.

For the time course experiment, bacterial solution with an OD of 3 were loaded on the grid in order to

be able to quantify early timepoints. Fluorescence intensity values for each square of the grids were

extracted with a custom Fiji imageJ (Schindelin et al., 2012) macro script and the maximum value for

each square was normalized to the highest measured fluorescence in all the conditions and replicates.

The data was plotted with R software (R Core Team, 2017).

Model derivation and parametrization

We model the expression dynamics of the TS by describing the change in time of the concentration of

the [LacI] and [TetR] as a balance between their regulated production and degradation. In addition, we

made the assumptions that the dynamics of the transcripts and promoter binding/unbinding is faster

than the dynamics production and degradation of the repressor proteins, and that the reporters follow

the same dynamics as their associated repressors.

The expression of LacI is regulated by TetR, which can inhibit the production of LacI by binding to the

TetO promoter. Modelling this interaction as a repressive Hill function we can describe the evolution in

time of LacI as, (1)

𝑑 𝐿𝑎𝑐𝐼

𝑑𝑡

𝛽𝑌1 𝐾𝑇𝑒𝑡𝑅 𝑇𝑒𝑡𝑅 𝑛𝑇𝑒𝑡𝑅

𝛿𝑌 𝐿𝑎𝑐𝐼

Where 𝛽𝑌 is the maximum production of LacI in the absence of the repressor TetR, KTetR the TetR

concentration producing half repression, nTetR the Hill coefficient and 𝛿𝑌 its degradation rate.

On the other hand, the expression of TetR is regulated at the same time by the repression of free LacI

in the system [LacIf] and the activation by AHL through the LuxR-AHL complex. Since the presence of

free [LacIf] is enough to silence the expression of [TetR] even in the presence of AHL, the production

can be modelled as the product of two Hill functions,

(2)

𝑑 𝑇𝑒𝑡𝑅

𝑑𝑡

𝛽𝑋1 𝐾𝐿𝑎𝑐𝐼 𝐿𝑎𝑐𝐼𝑓

𝑛𝐿𝑎𝑐𝐼 𝐾𝐴𝐻𝐿 𝐴𝐻𝐿

𝑛𝐴𝐻𝐿

1 𝐾𝐴𝐻𝐿 𝐴𝐻𝐿𝑛𝐴𝐻𝐿 𝛿𝑋 𝑇𝑒𝑡𝑅

Where 𝛽𝑋 is the production rate in absence of LacI and saturating amounts of AHL. The amount of free

LacI ([LacIf]), is controlled by IPTG, which can bind free LacI impeding the binding with the LuxLac

promoter,

(3)

𝐿𝑎𝑐𝐼𝑓 𝐿𝑎𝑐𝐼

1 𝐾𝐼𝑃𝑇𝐺 𝐼𝑃𝑇𝐺



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In order to parametrize the mathematical model Eqs. (1-3), we compared the experimentally observed

cellular states with the stable steady states of the theoretical dynamical system for different sets of

concentrations of AHL and IPTG. The experimental steady states were obtained by using the gated

expression in cellular populations (see Flow Cytometry for details). For each gated population, the state

was accepted when it contained at least 15% of the cellular population.

The stable states for the mathematical model ( 𝐿𝑎𝑐𝐼 ∗, 𝑇𝑒𝑡𝑅 ∗) were computed by solving Eqs. (1-3) at

equilibrium condition 0,

𝐿𝑎𝑐𝐼 ∗ 𝑓 𝑇𝑒𝑡𝑅 ∗ 𝛽 /𝛿1 𝐾 𝑇𝑒𝑡𝑅 ∗

𝑇𝑒𝑡𝑅 ∗ 𝑓 𝐿𝑎𝑐𝐼 ∗; 𝐼𝑃𝑇𝐺 , 𝐴𝐻𝐿 𝛽 /𝛿

1𝐾 𝐿𝑎𝑐𝐼 ∗

1 𝐾 𝐼𝑃𝑇𝐺

𝐾 𝐴𝐻𝐿1 𝐾 𝐴𝐻𝐿

Thus, finding the available steady states for a given condition is reduced to finding the roots of 𝐺 𝑥 in

the 1-dimensional equation 𝐺 𝐿𝑎𝑐𝐼 ∗ 𝑓 𝑓 𝐿𝑎𝑐𝐼 ∗; 𝐼𝑃𝑇𝐺 , 𝐴𝐻𝐿 𝐿𝑎𝑐𝐼 ∗ 0. This was done by

finding the number and approximate location of the roots by evaluating the sign of 𝐺 𝑥 over a

logarithmically spaced set along the possible values of 𝐿𝑎𝑐𝐼 ∗ 𝑓 , . All the values found were

refined by using the Brent-Dekker method with hyperbolic extrapolation. Finally, the stability of all the

possible found states was addressed by evaluating the eigenvalues of the diagonalized Jacobian

corresponding to Eqs (1-2).

In order to compare the computational steady states with fluorescence measurements we assumed a

linear relationship between fluorescence and gene expression,

𝐹 𝛼 𝜔 𝐿𝑎𝑐𝐼 ∗ , 𝐹 𝛼 𝜔 𝑇𝑒𝑡𝑅 ∗

Thus, the parametrization of the problem is reduced to the inference of 11 identifiable parameters 𝜃𝛼 ,𝛼 ,𝛽 ,𝛽 ,𝐾 ,𝐾 , ,𝐾 ,𝐾 ,𝑛 ,𝑛 ,𝑛 , where 𝛽 and 𝛽 summarize the parameter

products 𝛽 𝜔 𝛽 /𝛿 , and 𝛽 𝜔 𝛽 /𝛿

The ensemble of parameters in the mathematical model compatible with experimental observations was

inferred using Markov Chain Monte Carlo. In particular, we made use of Multiple-try Differential Evolution

Adaptative Metropolis algorithm (Laloy and Vrugt, 2012) using the PyDream implementation (Shockley

et al., 2018). The Likelihood function used to evaluate the goodness of a given set of parameters is

defined as,

ℒ 𝜃|data 𝑝 min𝐹 𝐹 ,

𝜎

𝐹 𝐹 ,

𝜎

Where the index 𝑖 in the sum runs over all the experimentally detected cell states for each experimental

condition. The experimental fluorescence (𝐹 ,𝐹 ), corresponds with the median of the gated

population of cells for the i-th observed state. Similarly, 𝜎 and 𝜎 are the standard deviation

of the gated populations of each state. The theoretical prediction for each parameter set 𝜃 and conditions



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20

of state 𝑖, is given by 𝐹 , and 𝐹 , , where the superindex 𝑗 presents the different theoretically

predicted stable states in the case of bistability. Finally, parameter 𝑝 penalizes that for the condition

given by state 𝑖 (concentrations of AHL and IPTG) the number of different states detected experimentally

is not the same than the number of stable states predicted in the mathematical model. Thus, 𝑝 promotes

parameter sets that match monostable and bistable regions between experimental and computational

steady states. In particular 𝑝 0 if the number of stable steady states match, and 𝑝 0 otherwise. For

the inference used in the manuscript, a value of 𝑝 10 was used. The inferred parameters are given

in Table 1.

Table 1: Parameters. Summary of values of parameters inferred from the experimental data

corresponding to the distributions of Figure 3. For each parameter, the median and the 95% credibility

interval of each marginal distribution are indicated. The median of each parameter corresponds with the

value used in the mathematical model for the rest of simulations of the manuscript.

Morphogen diffusion

The diffusion of the morphogens (AHL, IPTG, and aTc) on the agar plate is assumed to be a 2-D free

diffusion determining the concentration of each chemical 𝜌 𝑥,𝑦, 𝑡 at different positions of the plate. This

is modelled through the partial differential equations

𝜕𝜌𝜕𝑡

𝐷𝜕 𝜌𝜕𝑥

𝜕 𝜌𝜕𝑦

,

𝜕𝜌𝜕𝑡

𝐷𝜕 𝜌𝜕𝑥

𝜕 𝜌𝜕𝑦

.

Parameter Median Credibility Interval

𝛼 364 a.u. (252, 444) a.u.

𝛼 310 a.u. (153, 422) a.u.

𝛽 362 a.u. (200, 598) a.u.

𝛽 438 a.u. (322, 644) a.u.

𝐾 2.22 mM (1.04, 3.16) mM

𝐾 133 𝜇M (59.5, 318) 𝜇M

𝐾 4.17 10 a.u. (2.11, 8.90) 10 a. u.

𝐾 27.4 10 a.u. (6.12, 99.3) 10 a.u.

𝑛 2.17 (1.00, 3.73)

𝑛 1.61 (1.04,2.23)

𝑛 2.29 (1.16, 5.01)



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This equation was solved by discretizing the space using the square experimental grid. The boundary

conditions imposed by the experiment are given by the constant concentration of morphogen along one

side of the square 𝜌 0,𝑦, 𝑡 𝜌 , and assuming sinks in the rest of the perimeter of the

square𝜌 𝐿,𝑦, 𝑡 𝜌 𝑥, 0, 𝑡 𝜌 𝑥, 𝐿, 𝑡 0 , where L is the length of the side of the square. In order to

take into account the dilution effect of the initial concentration pipetted 𝑐 on the agar before stablishing

the gradient, we set 𝜌 𝑐 𝜉. Parameters used for the grid assays are 𝜉 10 and 𝐷 2.5 10 cm /h

(Basu et al., 2005, Miyamoto et al., 2018). Finally, in order to provide a time scale to the dynamics of

protein, not available from the MCMC fitting of the steady states of the system, the degradation rate of

the LacI and TetR was set to 𝛿 8.3 h ((Wu et al., 2011).

Boundary position in the grid model was computed by analyzing the red fluorescence levels along the

central strip of the grid 𝑇𝑒𝑇𝑅 𝑥, , 𝑡 . For each time point, a boundary was considered when the

difference in fluorescence at both sides of the gradient was above a certain threshold

𝑇𝑒𝑇𝑅ℓ

, , 𝑡 𝑇𝑒𝑇𝑅 𝐿ℓ

, , 𝑡 250, where ℓ is the distance between two adjacent cells of the

grid. The position of the boundary 𝑥 was set as the last cell of the grid were the concentration is above

the mean point of the fluorescence range for each given timepoint

𝑇𝑒𝑇𝑅ℓ

, , 𝑡 𝑇𝑒𝑇𝑅 𝐿ℓ

, , 𝑡 /2.

Acknowledgments

We thank Florence Gauye for excellent technical assistance and all Schaerli lab members for useful

discussions. This work was funded by Swiss National Science Foundation Grant 31003A_175608 and

an IPhD SystemsX.ch grant. RP-C acknowledges UCL Mathematics Clifford Fellowship.

Author Contributions

IB, RPC and YS designed research. IB performed experiments and analyzed data. RPC carried out the

mathematical modelling and the data fitting. IB, RPC and YS wrote the manuscript. All authors have

given approval to the final version of the manuscript.

Declaration of Interest

The authors declare no conflict of interest.



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